11 The classical limit

In the classical limit, the quantum-mechanical probability calculus, which assigns nontrivial probabilities (probabilities between 0 and 1) to measurement outcomes, degenerates into a probability calculus that assigns trivial probabilities (either 0 or 1) to measurement outcomes. In the classical theory, a probability algorithm based on the maximum of in principle available information is represented by a point P in a phase space, and measurement outcomes are represented by subsets of this space. Because P assigns trivial probabilities, it can be thought of as a state in the clas­si­cal sense: a col­lec­tion of pos­sessed prop­er­ties. We are licensed to believe that if the prob­a­bil­ity of find­ing a given property is 1, it is because the sys­tem pos­sesses this property, independent of measurements. In the quantum theory, a probability algorithm based on the maximum of in principle available information is represented by a 1-dimensional subspace L of a complex vector space, and measurement outcomes are represented by the subspaces of this space. Because L assigns nontrivial probabilities, it cannot be thought of as a state in the clas­si­cal sense.

Hence it may be said that the quantum laws, which correlate the probabilities of measurement outcomes probabilistically, degenerate into laws that correlate possessed properties deterministically. And because deterministic correlations between possessed properties lend themselves to causal interpretations, it may be said that the quantum-mechanical algorithms, which serve to compute probabilities of possible measurement outcomes on the basis of actual outcomes, degenerate into algorithms that serve to compute effects that the properties of material objects have on the properties of material objects. What they don’t is degenerate into (descriptions of) physical mechanisms or processes by which the properties of material objects act on the properties of material objects. As David Mermin[1] reminisced near the end of a distinguished career:

When I was an undergraduate learning classical electromagnetism, I was enchanted by the revelation that electromagnetic fields were real. Far from being a clever calculational device for how some charged particles push around other charged particles, they were just as real as the particles themselves, most dramatically in the form of electromagnetic waves, which have energy and momentum of their own and can propagate long after the source that gave rise to them has vanished.

That lovely vision of the reality of the classical electromagnetic field ended when I learned as a graduate student that what Maxwell’s equations actually describe are fields of operators on Hilbert space. Those operators are quantum fields, which most people agree are not real but merely spectacularly successful calculational devices. So real classical electromagnetic fields are nothing more (or less) than a simplification in a particular asymptotic regime (the classical limit) of a clever calculational device. In other words, classical electromagnetic fields are another clever calculational device.

In other words, Feynman’s assessment of quantum physics[2] — “There are no ‘wheels and gears’ beneath this analysis of Nature” — applies equally to classical physics. This should not come as a surprise. After all, in making the transition from quantum to classical we discard a considerable amount of information and, in so doing, sacrifice a great deal of explanatory power. What would be surprising is if classical physics could explain what quantum physics can’t explain.

Having said this, it shouldn’t be necessary to give another thought to what the approximate laws of classical physics tell us about the nature of Nature. Being approximate laws, they tell us nothing of the sort. As Feynman stressed at the beginning of his famous Caltech lectures,[3] “philosophically we are completely wrong with the approximate law” [original emphasis]. If it is nevertheless necessary to dwell on the “worldview” of classical physics, it is because the classical laws are all but universally taught before the quantum laws, and because along with the classical laws most student are force-fed a considerable amount of metaphysical embroidery that cannot but frustrate their later efforts to make sense of the quantum laws. Mermin[1] was able to rid himself of “our habit of inappropriately reifying our successful abstractions,” but how many physicists are? In answer to this question, it may not be inappropriate to quote philosopher of science Dennis Dieks:

As quantum theory is the reigning physical paradigm, one would expect that physicists have a quantum mechanical worldview in the back of their minds when pursuing their research. However, it has become obvious by now that that is not the case. Still, the need for a physical worldpicture makes itself felt. The reaction of most physicists is to substitute a kind of common-sense, quasi-classical, picture for the quantum mechanical one that really would be needed. Most physicists think of quantum objects as very small copies of everyday objects, and in effect use the conceptual scheme of classical physics. Of course, they know that something is wrong here, and that a consistent use of classical ideas will lead into trouble. But that problem rarely presents itself in an acute form. There is always the mathematical formalism of quantum mechanics with which calculations that lead to observational consequences are made; the pictures associated with the theory do not really intrude into those calculations. Because the mathematical formalism is consistent, no inconsistencies will be encountered when making predictions. Paradoxes and bewilderment only occur if one wonders about how the calculated and predicted experimental outcomes can be realized by natural processes.[4]